Abstract

In the semiclassical regime, we obtain a lower bound for the counting function of resonances corresponding to the perturbed periodic Schrödinger operator . Here is a periodic potential, a decreasing perturbation and a small positive constant.

1. Introduction

The quantum dynamics of a Bloch electron in a crystal subject to external electric field, which varies slowly on the scale of the crystal lattice, is governed by the Schrödinger equation
Here is periodic with respect to the crystal lattice , and it models the electric potential generated by the lattice of atoms in the crystal. The potential is a decreasing perturbation and a small positive constant.

There has been a growing interest in the rigorous study of the spectral properties of Bloch electrons in the presence of slowly varying external perturbations (see [1–11]).

Since the work of Peierls [10] and Slater [11], it is well known that, if is sufficiently small, then solutions of are governed by the “semiclassical” Hamiltonian
Here is one of the “band functions” describing the Floquet spectrum of the unperturbed Hamiltonian
One argues that for suitable wave packets, which are spread over many lattice spacings, the main effect of a periodic potential on the electron dynamics consists in changing the dispersion relation from the free kinetic energy to the modified kinetic energy given by the Bloch band.

The problem of resonances has been examined in [12] for the one-dimensional case and in [13] for the general case. In particular, a similar reduction to (1.2) for resonances has been obtained in [13].

This paper continues our previous works [13, 14] on the resonances and the eigenvalues counting function for . In [14], Dimassi and Zerzeri obtained a local trace formula for resonances. As a consequence, they obtained an upper bound for the number of resonances of in any -independent complex neighborhood of some energy . The purpose of this paper is to give a lower bound for the number of resonances of .

In the case where , it is known that, for in the analytic singular support (from now on for short) of the distribution , then the operator has at least resonances in any -independent complex neighborhood of (see, e.g., [15]). Here

Using the explicit formula of we see that the analytic singular support of the distributions and coincide.

In the case where the situation is different. Following Theorem 1.6 in [14] and Lemma 2.1 of the next section, we have to change by
which is the integrated density of states corresponding to the nonperturbed Hamiltonian (see Section 2).

If is a simple eigenvalue near some point , then is a smooth function, and if is a critical value, we expect in general that will belong to the analytic singular support of . In particular, we expect that near every point there exists at least , , resonances.

Multiple eigenvalues ) can also give rise to singularities of and then lead to the existence of resonances near .

The purpose of this paper is to describe all these situations. Some results of this paper are announced without proofs in [16].

The paper is organized as follows: in the next section, we introduce some notations and state some technical lemmas. In Section 3 we give an upper bound for resonances near singularities of the density of states measure generated by a band crossing. In Section 4 we give an upper bound for resonances near the edge of bands.

2. Preliminaries

Let be the lattice generated by the basis , . The dual lattice is defined as the lattice generated by the dual basis determined by , . Let be a fundamental domain for , and let be a fundamental domain for . If we identify opposite edges of (resp., ), then it becomes a flat torus denoted by (resp., ).

Let be a real valued potential, and -periodic. For in , we define
as an unbounded operator on with domain . The Hamiltonian is semibounded and self-adjoint. Since the resolvent of is compact, the resolvent of is also compact, and therefore has a complete set of (normalized) eigenfunctions , , called Bloch functions. The corresponding eigenvalues accumulate at infinity, and we enumerate them according to their multiplicities:
Since , the band function is periodic with respect to . The function is called a band function, and the closed intervals are called bands.

Standard perturbation theory shows that is a continuous function of and is real analytic in a neighborhood of any such that
We fix in the spectrum of the unperturbed operator . We make the following hypothesis on the spectrum of the unperturbed Schrödinger operator. (H1) For all with , the eigenvalue is simple and .

Now, let us recall some well-known facts about the density of states associated with . The density of states measure is defined as follows:
where is a fundamental domain of . Since the spectrum of is absolutely continuous, the measure is absolutely continuous with respect to the Lebesgue measure . Thus, the density of states of , is locally integrable.

We now consider the perturbed periodic Schrödinger operator:
where . We assume that there exist positive constants and such that extends analytically to and
where . Here , denote, respectively, the real part and the imaginary part of .

This assumption allows us to define the resonances of by the spectral deformation method (see [17]). We follow essentially the presentation of [13].

Let be -periodic. For , we introduce the spectral deformation family defined by for all
where and its Jacobian. Here is the semiclassical Fourier transform:
Consider, for , the family of unitarily equivalent operators
It was established in [13, Proposition 2.8] that extends to an analytic type- family of operators on with domain . Moreover, under the assumptions (H1) and (2.6), there exists a neighborhood of and a small positive constant such that, for with , the spectrum of in consists of discrete eigenvalues of finite multiplicities that lie in the lower half plane (see [13, formula (4.9)]). These eigenvalues are -independent under small variations of and are called resonances. We will denote the set of resonances by .

For , we set
For , let
Similarly, for , we set
Clearly, (resp., ) is a decreasing function of (resp., an increasing function of ) and

Lemma 2.1. The distributions and are real valued of order ≤1. Moreover, in, one has

Proof. Applying Taylor’s formula to the right-hand side of (2.10), we obtain
which together with (2.6) imply that is a distribution of order ≤1, with
Consequently, is well defined in and for all , we have
This ends the proof of the lemma.

Let be an open-bounded set in , and let be a complex neighborhood of . Let be analytic on and real valued for all in . Let us introduce the real function
For , we set

Lemma 2.2. Let , and let , be as above. One assumes that (i) for all ,(ii). Then the function
is analytic near .

Proof. Let be a small positive constant such that when . Without any loss of generality we may assume that for all . By the change of variable we have
Clearly the right-hand side of the above equality is analytic. Combining this with the fact that is constant for near we get the lemma.

Lemma 2.3. If has a nondegenerate extremum at with and if for all , then
where and are analytic near zero and
Here is the Heaviside function and corresponds to a minimum (maximum, resp.).

Proof. Here we only give a sketch of the proof. For the details we refer to [18]. Without any loss of generality, we only consider the case of minimum. By Morse lemma there exist a neighborhood of , and a local analytic diffeomorphism such that
By a simple calculus we show, using polar coordinates, that the integral of the r.h.s. is equal to . On the other hand, since for , it follows from Lemma 2.2 that
is analytic near . This ends the proof of the lemma.

3. Lower-Bound Near Singularities due to Band Crossing

Here we are interested in the singular support (which will be denoted by ). Recall that if and only if is near . The case of analytic singular support can be treated similarly.

In this section we study resonances near singularities of generated by a band crossing. We will only consider the two-dimensional case. With similar assumptions, one can treat the case .

We assume that is double eigenvalues and that for all such that , is simple and .

Since is analytic in , this implies that, for (with small enough), the span , of the eigenvectors of corresponding to eigenvalues in the set , has a basis , , which is orthonormal and real analytic in . The restriction of to has the matrix
which can be written
where , , and are real valued. Next the periodic potential is assumed to have the symmetry . This symmetry is typical of metals. This symmetry forces to be real valued (i.e., ), (see [19]). Consequently, near we have
We assume that , are independent. Since , is a basis in . Set .

Lemma 3.1. Let be as above. One assumes that
Then there exist an open connected neighborhood of and analytic functions and such that
with

Proof. To simplify the notation we assume that and .Let be a neighborhood of . We introduce
so that
Due to Lemma 2.2, the right-hand side of the above equalities is analytic near 0.Since , are independent, there exist a neighborhood of , and a local analytic diffeomorphism such that, with the change of variable , we obtain
where and are analytic near and . Using polar coordinates and making the change , in the second integral, we get
which can be written
where . Since
uniformly on , there exist (independent on ) such that from into is an analytic diffeomorphism. Hence, for small enough
where
Using that
we deduce .

We denote by the number of elements of , counted with their multiplicity. The main result of this section is the following.

Theorem 3.2. Let with . One assumes the following. (i)The periodic potential satisfies .(ii)There exists such that .(iii)For all such that , the eigenvalue is simple and .(iv)The numbers satisfy (3.4), and . Here is the interval given by Lemma 3.1.(v) satisfies (H1).Then for all -independent complex neighborhoods of , there exist sufficiently small and such that, for ,

Proof. Without any loss of generality we may assume that . Set
where is the function given in Lemma 3.1.The assumption that ensures that, in the study of near , one only needs the value of in given by (3.4). More precisely, it implies that
for near .Since is smooth, the first term of the right-hand side of the above equation is also smooth.Clearly, it follows from assumption (2.6) and Lemma 2.2 that the is a discrete set. Thus, the point is isolated in . We recall that we have assumed that .Let (resp., be equal to one near zero (resp., ). Here is the disc of center and radius . Set and . We choose small enough such that
To study the second term of the right-hand side of (3.18), we write it in the form
Since is smooth the term (3) is also smooth. Using (3.19) and the fact that the support of is small for , we see that the term (5) is near .Now, we claim that
First, from a standard result on the singular support, we have
Consequently, to prove the claim it suffices to show that . We recall that has a compact support.A simple calculus and Lemma 3.1 yield
Here is the Fourier transform of . Consequently, if and only if , where is the Schwartz space of function of rapid decrease.On the other hand, (3.19) implies that . Combining this with the above remarks we get the claim.Summing up, we have proved that .Now, applying the following result of [14] we obtain Theorem 3.2.Theorem 3.3 (see [14]). Let . Assume that satisfies (H1). Then for every -independent complex neighborhood of , there exists sufficiently small and large enough such that, for ,

Remark 3.4. Let be a singularity of the integrated density of states, generated by a band crossing. Theorem 3.2 shows that there is at least resonances near , where is in the singular support of the distribution defined by

4. Lower Bound of the Counting Function near the Edges of Bands

In this section we study resonances generated by analytic singularities of near the edge of bands. The following result is a consequence of Lemma 2.3.

Lemma 4.1. Let . One assumes the following. (i)If , then is a simple eigenvalue of .(ii)There exist and such that , , and .(iii)For all with , .Then there exists an open connected neighborhood of such that
where and are analytic near zero and . Here, corresponds to a local minimum (maximum, resp.).

Now, repeating the arguments in the proof of Theorem 3.2 and using Lemma 4.1, we obtain the following.

Theorem 4.2. Let with . One assumes the following. (i) satisfies (H1),(ii) satisfies the assumptions of Lemma 4.1,(iii). Here is the interval given by Lemma 4.1.Then for all -independent complex neighborhoods of , there exist sufficiently small and such that, for ,

Remark 4.3. Notice that the assumptions (iv) in Theorem 3.2 and (iii) in Theorem 4.2 are satisfied if is small.